Model session times and comments with exponential/Poisson
Overview
This question evaluates understanding of memoryless processes and count-data modeling, covering exponential (constant-hazard) distributions for session durations and Poisson approximations for comment counts within survival analysis and count-data modeling in Statistics & Math for a Data Scientist role.
Session Duration Memoryless Assumption and Poisson Comment Counts
Setup
- We model user session end times with a constant hazard (memoryless) over time.
- Separately, for commenting behavior: each time a user views a post, they independently leave a comment with small probability p; a user views M posts in total.
Tasks
(a) Under a constant hazard assumption, derive the implied distribution of session durations and state the memoryless property explicitly.
(b) Describe two empirical checks, using survival plots or hazard estimates, that can be used to validate the constant-hazard (memoryless) assumption in session duration data.
(c) Justify a Poisson model for the per-user comment count. Specify its parameter in terms of p and M, and state the conditions under which the approximation is accurate.
(d) Describe diagnostics that would suggest overdispersion or zero inflation in comment counts and name an alternative model you would consider in each case.
Solution
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